Análisis de la traza compleja

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Es encontrar la representación en número complejo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)} de una serie de tiempo real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)= f(t)+ i f_{\perp}(t)=A(t)e^{i \phi(t)} } ,

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{\perp}(t) } es la es la serie en cuadratura, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(t)} es la amplitud de la envolvente de la traza (también llamada fuerza de la reflexión), y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)} es la fase instantánea. Despliegues de la fase instantánea (o coseno de la fase instantánea) muestra la continuidad de un evento. La frecuencia instantánea es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\phi(t)/dt} . La frecuencia instantánea puede ser considerada como la frecuencia de la sinusoide compleja que localmente ajusta mejor a una traza compleja. Es empleado para determinar atributos sísmicos. En el dominio del espacio, "local" también se emplea en lugar de “instantáneo”. Ver Figura C-11 y Taner et al. (1979). Los análisis de traza compleja comúnmente involucran la transformada de Hilbert .

FIG. C-11. Análisis de traza compleja. Traza real (a) y de cuadratura (b) para una parte de un registro sísmico. La envolvente se muestra con la línea punteada en (a) y (b). La fase instantánea se representa en (c), la frecuencia instantánea en (d) y la frecuencia promedio ponderada como la curva punteada en (d). (e) Diagrama isométrico de una traza compleja. (De Taner et al.,1979.)


Fundamentos matemáticos del análisis de la traza compleja

La noción de una traza compleja o analítica comienza con el resultado más general de que cualquier función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z)= \mbox{Re } F(z) + i \mbox{Im } F(z) } , donde math> z = x + i y </math> obedece la relación que (la cual se deriva de la fórmula integral de Cauchy) que Spiegel (1964) [1] o Levinson y Redheffer (1970). [2] señalan.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Im } F(z) = {\mathcal H} \left[ \mbox{Re }F(z) \right] }


donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathcal H} } es una operación conocida como la transformada de Hilbert, en cualquier región donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) } es una función analítica. ("Analítico" significa que la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dF/dz } existe)


Amplitud instantánea

Podemos escribir (empleando la relación de Euler)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) = A(z)\exp(i \phi(z)) = A(z)\left[ \cos(\phi(z) ) + i \sin(\phi(z)) \right]}

donde el módulo es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(z) = \sqrt{ (\mbox{Re } F(z))^2 + i (\mbox{Im } F(z))^2 } }



Fase instantánea

La fase es entonces, el arco tangente del cociente de las partes imaginaria y real

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(z) = \arctan \left( \frac{\mbox{Im } F(z)}{\mbox{Re } F(z)} \right) } .

Por lo tanto, la parte real de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) } es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Re } F(z) = A(z) \cos(\phi(z)) }

y la parte imaginaria es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Im } F(z)) = A(z) \sin(\phi(z))} .


La traza Compleja (o Analítica)

Consideremos ahora la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t)} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} incrementa monotónicamente. La función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t)} describe una curva en el volumen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x(t) ,y(t) ,t)} . Esta curva posee un solo valor en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} para este volumen, obteniendo así la siguiente parametrización

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z(t)) = A(z(t)) \left[ \cos(\phi(z(t))) + i \sin(\phi(z(t))) \right] }

Porque Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t) } es un solo valor en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } dentro del volumen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x,y,t)} podemos escribir, sin pérdida de generalidad


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t) = A(t)\left[\cos (\phi(t)) + i \sin(\phi(t)) \right] }


la cual describe una hélice en el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } , definida por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (0,0,t)} en el volumen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x,y,t)} .


Ahora afirmamos que nuestros datos registrados Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)} son la parte real de esta traza compleja Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)} , por lo tanto:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t) = A(t) \cos (\phi(t)) }


y la parte imaginaria, o también llamada "traza en cuadratura" es


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{\perp} (t) = A(t) \sin (\phi(t)) } .


El módulo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(t)} es la "amplitud instantánea", también conocida como la "función envolvente" de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t) } . La función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)} es conocida como la "fase instantánea" de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)} .



Frecuencia instantánea

Una "frecuencia instantánea" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t)} puede ser definida como la velocidad de cambio de la fase instantánea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t) = \frac{d\phi(t)}{dt} } .

.

Computacionalmente, la fase instantánea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi } calculada de este modo puede ser "envuelta", es decir, puede tener saltos hasta de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\pi } , debido al hecho de que el cálculo numérico computacional de la función arcotangente está limitado para el rango principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\pi < \phi \le \pi } .

Es preferible expresar la función arcotangente en su forma diferencial, para evitar problemas de envolvimiento de la fase


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t) = \frac{d\phi(t)}{dt} = \frac{d}{dt} \arctan \left( \frac{\mbox{Im } F(t)}{\mbox{Re } F(t)} \right) = \frac{ (\mbox{Im } F(t))^\prime \; (\mbox{Re } F(t)) - (\mbox{Im } F(t)) \; (\mbox{Re } F(t)) ^\prime}{(\mbox{Re } F(t))^2 + (\mbox{Im } F(t))^2} }


donde reconocemos que el denominador es el cuadrado de la amplitud instantánea.

Esta formulación del análisis de la traza compleja, introducida a la comunidad geofísica por Taner, Koehler y Sheriff (1979) [3], ha encontrado una vasta aplicación en el procesamiento sísmico para la interpretación

Desde 1979, una colección de los llamados atributos sísmicos han sido creados.


Referencias

  1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
  2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
  3. M. T. Taner, F. Koehler, and R. E. Sheriff (1979). "Complex seismic trace analysis." GEOPHYSICS, 44(6), 1041-1063. doi: 10.1190/1.1440994


Vínculos externos

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